Buffy the Vampire Slayer

When Spike gets shot by the Initiative soldiers with the tracer, he's wearing his coat, but when he goes to Giles and the gang for help, there's no hole or damage to his coat.
In fact, he has to partially remove his coat to show them the tracer (which did leave a hole in his shirt).

Somehow, near the end of the episode Buffy gets into Giles' apartment without anyone noticing her until she speaks up. Even if she did so as quietly as possible it's bright sunshine outside, the room should have lit up when Buffy opened the door.

When Anya is seen in closeup while playing poker a small round clock can be seen on a table next to the phone. Although the scene lasts about two minutes, the time on the clock stays at 8:40.

In the poker scene the amount of potato chips in the blue bowl keeps changing, even though no one eats any of them.

Xander and Anya are trying to sell Giles some Boost Bars. When all three are seen together Anya is sitting with her legs tucked under her. When just Xander and Anya are shown Anya hasn't moved, yet her skirt has moved and her right arm is resting on a pillow that isn't in the other shot.

When Professor Walsh introduces Dr. Angleman there is a technician on his right lowering his arms to his side. In the very next shot the technician has his arms crossed and is twirling a pen in his right hand.

In The Witch, Willow called herself, Xander and Giles the 'Slayerettes'. This term wasn't used again until the episode The I in Team.

At the end of the early crypt scene where Giles repays Spike, Giles opens the door to leave and Spike moves to the side to avoid the sunlight. However, throughout the scene Spike does not seem to mind the large amount of direct sunlight coming through the window. And if we say that the sunlight coming through the window isn't direct enough, then it follows that the sunlight coming through the door wouldn't be either, unless it's early morning or late evening and the door just happens to face directly east or west, as appropriate. It seems unlikely a vampire would choose a place with that kind of vulnerability.

There is a guy on the ground in the teaser who disappears in later shots.

When Maggie enters the 314 door you first see it close from the outside, but when the shot changes you can see her finish closing the door, when it already finished closing in the previous shot.

When the Initiative boys attack Spike, he shoves one down and throws another one into a tree, hard, but the chip obviously doesn't respond, possibly because no significant harm was being done, but this needed to be highlighted or explained better.